Abraham is a "righteous" (fair and honest) man
He insists on paying because of his honorable culture heritage that he was likely raised with and that he passes on to all future generations. This also helps to set the standard of "fairness" for the rest of the Bible.
While the word could be translated "sell", the context clarifies that this is the kind of transaction intended to be free of charge; it should be translated "give [as a free gift]". Yes, studying the original languages works that way, as does context clarify one of many possible definitions in English.
It works the same in English. Consider: "I'll give it to you for $45."
Behind the "why"
An important question is about why the field was offered free of charge in the first place. That should have been a normal courtesy, given the circumstances of Abraham as a foreigner, not to mention that they wanted favor and good relations with Abraham.
When he begs to be allowed to pay so he can bury his wife, he basically means, "I won't accept it without paying, I have made up my mind, please don't argue with me, accept my money, that way I can get on with my wife's funeral, which is more important to me."
What "right[eous]ness" means
We could say that Abraham's desire to pay relates to "personal honor", but such personal honor must include having concern for the well being of others. Receiving a field for no money may be generous from the giver, but it is not "fair behavior" on Abraham's part because then the giver would be worse off without compensation. Abraham is demonstrating a value in the Bible, seen cover to cover, of "being fair" to others.
The Bible's concept of "righteousness" (AKA 'fairness') begins with the practical things
We could say that we do justice on Earth to reflect God's justice in Heaven, but that gets off the point here, that righteousness must include being fair in the practical matters.
This is a basic "fairness" or "righteousness", the practical kind, not a Church theology spiritually-emphasized version of "righteousness" that developed later in history. God demands "righteousness" with practical matters in many passages of Scripture and it applies on many levels...
Leviticus 19:35 (NASB95)
You shall do no wrong in judgment, in measurement of weight, or capacity.
Deuteronomy 25:13–16 (NASB95)
13“You shall not have in your bag differing weights, a large and a small.
14“You shall not have in your house differing measures, a large and a small.
15“You shall have a full and just weight; you shall have a full and just measure, that your days may be prolonged in the land which the Lord your God gives you.
16“For everyone who does these things, everyone who acts unjustly is an abomination to the Lord your God.
Proverbs 20:23 (NASB95) Similar to Proverbs 11:1
Differing weights are an abomination to the Lord,
And a false scale is not good.
We also see this theme of "paying a fair price" in terms of wages throughout the Bible, as an extended application of the same ethical and hoborable conduct. Consider...
Leviticus 19:13 (NASB95)
You shall not oppress your neighbor, nor rob him. The wages of a hired man are not to remain with you all night until morning.
Deuteronomy 24:15 (NASB95)
You shall give him his wages on his day before the sun sets, for he is poor and sets his heart on it; so that he will not cry against you to the Lord and it become sin in you.
Matthew 10:10b (NASB95)
for the worker is worthy of his support.
1 Timothy 5:18 (NASB95)
For the Scripture says, “You shall not muzzle the ox while he is threshing,” and “The laborer is worthy of his wages.”
"[Spiritual] righteousness" was a concept that developed atop this practical framework. We first learn from the Bible that "believing God" is a kind of "equal fairness", using Abraham's good example as the first demonstration.
Genesis 15:6 (NASB95)
Then he believed in the Lord; and He reckoned it to him as righteousness.
Insisting on paying for the field was merely a matter of expected conduct in Abraham's life, as his life helps set the tone for "fairness" in the Bible.
In fact, this passage can therefore be used to interpert other passages in the Bible that relate to justice, including all the examples given here and many more.
Note that this does not mean that Abraham is "perfect", but that he tries to practise a life of fair and equal standards.